Best Known (10−9, 10, s)-Nets in Base 5
(10−9, 10, 10)-Net over F5 — Constructive and digital
Digital (1, 10, 10)-net over F5, using
- net from sequence [i] based on digital (1, 9)-sequence over F5, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F5 with g(F) = 1 and N(F) ≥ 10, using
- a shift-net [i]
(10−9, 10, 11)-Net over F5 — Upper bound on s (digital)
There is no digital (1, 10, 12)-net over F5, because
- 3 times m-reduction [i] would yield digital (1, 7, 12)-net over F5, but
- extracting embedded orthogonal array [i] would yield linear OA(57, 12, F5, 6) (dual of [12, 5, 7]-code), but
(10−9, 10, 13)-Net in Base 5 — Upper bound on s
There is no (1, 10, 14)-net in base 5, because
- extracting embedded orthogonal array [i] would yield OA(510, 14, S5, 9), but
- the linear programming bound shows that M ≥ 419 921875 / 39 > 510 [i]